GiniIndex

The size plus 1 choose 2, less the sum over the columns of the column length plus 1 choose 2.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001541via IntegerPartition✓↗

GiniIndex(partition)The size plus 1 choose 2, less the sum over the columns of the column length plus 1 choose 2.

Typed signatures

Type	Package	Applies
(integer_partition) -> number	combinatorics	always

Domain: Partition statistics

Details

• Defined over IntegerPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerPartition and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Sum(_x) * (Sum(_x) + 1) / 2 - Sum(Map(
                                                                Function(i * (i +
                                                                              1) /
                                                                         2, i),
                                                                Map(
                                                                     Function(Count(Filter(
                                                                                            Range(1, Length(_x)),
                                                                                            Function(_x[j] >= i, j),
                                                                                          )), i),
                                                                     Range(
                                                                            1,
                                                                            Max(_x),
                                                                          ),
                                                                   ),
                                                              ))

Examples

Basic #

the partition $4 + 2 + 1$ of $7$

live ↗

In$\mathrm{GiniIndex}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$17$

the partition $3 + 3 + 1$ of $7$

live ↗

In$\mathrm{GiniIndex}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$16$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([1]), \text{GiniIndex})$Out$0$

Known value · FindStat St001541

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{GiniIndex})$Out$1$

Known value · FindStat St001541

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{GiniIndex})$Out$3$

Known value · FindStat St001541

About

Examples	5
Crosswalk	FindStat